FRG: Collaborative Research: Arithmetic and equidistribution on homogeneous spaces

FRG：协作研究：齐次空间上的算术和等分布

• 批准号: 0903110
• 负责人: Akshay Venkatesh
• 金额: $ 8.35万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0903110&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Arithmetic; equidistribution

In recent years, it has become clear that many interesting problems, in particular problems in arithmetic, quantum chaos and the theory of L-functions, may be profitably reduced to questions concerning equidistribution of points or measures on homogeneous spaces. These questions regarding equidistribution can be approached from many angles. Two theories which have proved to be particularly well-suited to study such questions are the spectral theory of automorphic forms, which is closely related to the theory of L-functions, and the theory of dynamical systems, particularly the study of unipotent and more general flows on these homogeneous spaces. Recently there has been considerable progress involving tools such as special value formulae for L-functions, and (partial) classification results for measures invariant under higher rank torus actions. Particularly exciting is the possibility, already realized in some instances, of combining these techniques. The purpose of the proposed FRG is to investigate further this circle of ideas, which we believe has the potential to impact many other problems related to the above. The result of these investigations will be a deeper understanding of the dynamics of group actions on homogeneous spaces, of the analytic theory of automorphic forms, and the (sometimes unexpected) applications to problems ofarithmetic nature.The present project is concerned with a surprising link between two classical fields of mathematics of quite disparate origin: number theory and dynamics. The study of number theory began thousands of years ago, motivated, in significant part, by questions about prime numbers. On the other hand, ergodic theory and dynamics are mathematical fields of more recent provenance, which arose from studying the long-term evolution of complicated deterministic processes -- such as planetary motion. It is a striking fact (which has only recently begun to be heavily exploited) that, in certain contexts, ideas from ergodic theory interact very deeply with classical problems in number theory. This project will enhance our understanding of this inter-relation and how we can combine knowledge from both of these fruitful disciplines effectively.

近年来，许多有趣的问题，特别是算术、量子混沌和l函数理论中的问题，都可以有效地简化为齐次空间上点或测度的等分布问题。这些关于均分的问题可以从多个角度来探讨。两个已被证明特别适合研究这类问题的理论是自同构形式的谱理论，它与l-函数理论密切相关，以及动力系统理论，特别是研究这些齐次空间上的单幂和更一般的流。近年来，在l函数的特殊值公式和高阶环面作用下测度不变性的（部分）分类结果等工具方面取得了相当大的进展。特别令人兴奋的是，在某些情况下已经实现了将这些技术结合起来的可能性。拟议的FRG的目的是进一步调查这一系列想法，我们认为这些想法有可能影响与上述有关的许多其他问题。这些研究的结果将是对齐次空间上群作用的动力学，自同构形式的解析理论，以及（有时意想不到的）算术性质问题的应用有更深的理解。本项目关注的是两个起源完全不同的经典数学领域之间的惊人联系：数论和动力学。数论的研究始于几千年前，在很大程度上是由质数问题所激发的。另一方面，遍历理论和动力学是近代兴起的数学领域，起源于对复杂确定性过程（如行星运动）长期演化的研究。这是一个惊人的事实（直到最近才开始被大量利用），在某些情况下，遍历理论的思想与数论中的经典问题有着非常深刻的相互作用。这个项目将增强我们对这种相互关系的理解，以及我们如何有效地将这两个富有成果的学科的知识结合起来。